1 Benoit Boulet, Ph.D., Eng. Industrial Automation Lab McGill Centre for Intelligent Machines...

1

Benoit Boulet, Ph.D., Eng.Industrial Automation LabMcGill Centre for Intelligent MachinesDepartment of Electrical and Computer EngineeringMcGill University, Montreal

Necessary conditions for consistency Necessary conditions for consistency of noise-free, closed-loop frequency-of noise-free, closed-loop frequency-response data with coprime factor response data with coprime factor modelsmodels

American Control ConferenceJune 29, 2000

Benoit Boulet, June 29, 2000 2

OutlineOutline

1- Motivation: model validation for robust 1- Motivation: model validation for robust control control

2- Coprime factor plant models2- Coprime factor plant models

3- Consistency of closed-loop freq. resp. (FR) data with 3- Consistency of closed-loop freq. resp. (FR) data with coprime factor modelscoprime factor models

4- Example: Daisy LFSS testbed4- Example: Daisy LFSS testbed

5- Conclusion5- Conclusion

/ H


1- Motivation: model validation 1- Motivation: model validation for robust controlfor robust control

K GController

Plant

+

+

+

+

+

-

reference

Uncertainty

Output dist./Meas. noise

Meas. output

Input Dist.

• Typical feedback control systemTypical feedback control system

/ H


Design stabilizing LTI K Design stabilizing LTI K such that CL system is such that CL system is stablestable

““robust stability”robust stability”where is the space of stable where is the space of stable transfer functions,transfer functions,

and is a bound on the uncertainty:and is a bound on the uncertainty:

Robust control objectiveRobust control objective

P

K

z

y u

w1, 1r

H

: sup ( )Q Q j

H

r( ) ( )j r j

Motivation: model validation for robust control/ H


robust controlrobust control

P

K

z

y u

wCondition for robust Condition for robust stability as given by stability as given by small-gain theorem:small-gain theorem:

1closed-loop stable , 1r H

iff , 1Lr P KF




P

K

z

y u

w

Condition for robust Condition for robust stabilitystability

provides the provides the motivation to make motivation to make (the uncertainty) (the uncertainty) as small as possible as small as possible through better through better modelingmodeling

, 1Lr P KF

( )r j




P

K

z

y u

w

Conclusion:Conclusion:

Robust stability is Robust stability is easier to achieve if the easier to achieve if the size of the uncertainty size of the uncertainty is small.is small.

Same conclusion for Same conclusion for robust performance robust performance ( -( -synthesis)synthesis)



……uncertainty modeling is key to uncertainty modeling is key to good controlgood control

• From first principles: Identify From first principles: Identify nominal values of uncertain gains, nominal values of uncertain gains, time delays, time constants, high time delays, time constants, high freq. dynamics, etc. and bounds on freq. dynamics, etc. and bounds on their perturbationstheir perturbationse.g., e.g., ++, |, ||<b|<b

• From experimental I/O dataFrom experimental I/O data



2- Coprime factor plant models2- Coprime factor plant models

1:p p pG CM N J

• Perturbed left-coprime factorizationPerturbed left-coprime factorization

, ,, , ,

: , :

n n n m m m p pM N

p M p N

M N J C

M M N N

RH RH RH RH

Coprime factor plant models

wherewhere


Aerospace example: DaisyAerospace example: Daisy

• Daisy is a large flexible space structure Daisy is a large flexible space structure emulator at Univ. of Toronto Institute for emulator at Univ. of Toronto Institute for Aerospace Studies (46th-order model)Aerospace Studies (46th-order model)

1

23 23

23 23

p p p

p

p

G CM N J

M

N

RH

RH



definedefine

: N M RH

1

where bounds the size of the factor uncertainty: ( ) ( ),

: : 1r

r j r j

r

D RH

• Factor perturbationFactor perturbation

• Uncertainty setUncertainty set

• Family of perturbed plants Family of perturbed plants

: :p rG P D



block diagram of open-loop block diagram of open-loop perturbed LCFperturbed LCF

1where factor uncertainty is normalized such that,

( ) 1 ( ) ( ),j j r j

r

JNM

rI

C

( )P s



Block diagram of closed-loop Block diagram of closed-loop perturbed LCFperturbed LCF

NM

rI

C

1K

2K

( )H s

1 2assumption: , internally stabilize the plant

and provide sufficient damping for FR measurement

K K



3- Consistency of closed-loop 3- Consistency of closed-loop frequency-response data with frequency-response data with coprime factor modelscoprime factor models

• Model/data consistency problem:Model/data consistency problem:

Given noise-free, (open-loop,closed-Given noise-free, (open-loop,closed-loop) frequency-response data loop) frequency-response data obtained at frequencies , obtained at frequencies , could the data have been produced could the data have been produced by at least one plant model in by at least one plant model in ? ?

1

N p mi i

1, , N

:p rG P D


open-loopopen-loop model/data model/data consistency problem solved in:consistency problem solved in:

• J. Chen, IEEE T-AC 42(6) June 1997 (general J. Chen, IEEE T-AC 42(6) June 1997 (general solution for uncertainty in LFT form)solution for uncertainty in LFT form)

• B. Boulet and B.A. Francis, IEEE T-AC 43(12) B. Boulet and B.A. Francis, IEEE T-AC 43(12) Dec. 1998 (coprime factor models)Dec. 1998 (coprime factor models)

• R. Smith and J.C. Doyle, IEEE T-AC 37(7) Jul. R. Smith and J.C. Doyle, IEEE T-AC 37(7) Jul. 1992 (uncertainty in LFT form, optimization 1992 (uncertainty in LFT form, optimization approach) approach)

Consistency of closed-loop FR data with coprime factor models


closed-loop FR data caseclosed-loop FR data case

i

iNiM

ir I

iC

i

I

0

consistency equation at frequency i

where : ( ),

, 0

i i

U i i i

H H j

H

F

iH

1iK

2iK



Lemma 1Lemma 1

1

, 0

,

U i i i

L i i i

H

rank I H n

F

F

Lemma 2Lemma 2 (Schmidt-Mirsky Theorem) (Schmidt-Mirsky Theorem)

11 1inf : rank , ,i L i i i p L i iI H n H

F F



Lemma 3Lemma 3 (consistency at ) (consistency at )

( )

11

, 1

such that , 0

, 1

n n pi i

U i i i

p L i i

H

H

F

F

i



TheoremTheorem (consistency with CL FR (consistency with CL FR data)data)

1

11

consistent with LCF model only if

, 1, 1, ,

N

i i

p L i iH i N

F

Proof Proof (using boundary interpolation theorem)(using boundary interpolation theorem)

11

( ) such that ( )

only if

, 1, 1, ,

r i i i

p L i i i

s j r

H i N

D

F



This condition is This condition is not sufficient.not sufficient.

For sufficiency, the perturbation For sufficiency, the perturbation would have to be shown to stabilize would have to be shown to stabilize to account for the fact that the closed-to account for the fact that the closed-loop system was stable with the loop system was stable with the original controller(s)original controller(s)

We can’t just We can’t just assumeassume this a priori as it this a priori as it would mean that the original would mean that the original controller(s) is already robust!controller(s) is already robust!

1 2( ), ( )K s K s

( ) rs D( )H s



Example: Daisy LFSS testbedExample: Daisy LFSS testbed

• Nominal factorizationNominal factorization

• Bound on factor uncertaintyBound on factor uncertainty

• one of the plants in family of perturbed one of the plants in family of perturbed plants plants was chosen to be the actual plant generating was chosen to be the actual plant generating the 50 closed-loop FR data pointsthe 50 closed-loop FR data points

• 23 first-order decentralized SISO lead controllers 23 first-order decentralized SISO lead controllers were used as the original controllerwere used as the original controller

1

23 23 23 23, , ,

G CM NJ

M N J C

RH

0.001 1.414( )

2.32 1

sr s

s

,aG

P

1K

50

1i i

Example: Daisy LFSS Testbed


Example (continued)Example (continued)

i

iNiM

ir I

iC

i

I

0

consistency equation at frequency i

where : ( ),

, 0

i i

U i i i

H H j

H

F

iH

1iK

iJ



Example (continued)Example (continued)

• Model/data consistency check:Model/data consistency check:

11,p L i iH

F



5- Conclusion5- Conclusion

• Necessary condition for consistency of noise-Necessary condition for consistency of noise-free FR data with uncertain MIMO coprime factor free FR data with uncertain MIMO coprime factor plant model involves the computation of at plant model involves the computation of at the measurement frequenciesthe measurement frequencies

• Bound on factor uncertainty can be Bound on factor uncertainty can be reshaped to account for all FR measurementsreshaped to account for all FR measurements

• Sufficiency of the condition is difficult to obtain Sufficiency of the condition is difficult to obtain as one would have to prove that the factor as one would have to prove that the factor perturbationperturbation , proven to exist by the boundary , proven to exist by the boundary interpolation theorem, also stabilizes the interpolation theorem, also stabilizes the nominal closed-loop system.nominal closed-loop system.

p

( )r s

( ) rs D

N

25

Thank you!Thank you!

1 Benoit Boulet, Ph.D., Eng. Industrial Automation Lab McGill Centre for Intelligent Machines...

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